Equivariant resolutions over Veronese rings

Working in a polynomial ring $S=\mathbf{k}[x_1,\dots ,x_n]$ where $\mathbf{k}$ is an arbitrary commutative ring with $1$, we consider the $d^{th}$ Veronese subalgebras $R=S^{(d)}$, as well as natural $R$-submodules $M=S^{(\ge r,d)}$ inside $S$. We develop and use characteristic-free theory of Schur functors associated to ribbon skew diagrams as a tool to construct simple $G{L}_n(\mathbf{k})$-equivariant minimal free $R$-resolutions for the quotient ring $\mathbf{k}=R\mathrm{/}{R}_{+}$ and for these modules $M$. These also lead to elegant descriptions of ${\mathrm{T}\mathrm{o}\mathrm{r}}_i^R(M,M^{\mathrm{\prime }})$ for all $i$ and ${\mathrm{H}\mathrm{o}\mathrm{m}}_R(M,M^{\mathrm{\prime }})$ for any pair of these modules $M,M^{\mathrm{\prime }}$.